account the photon polarization. The excitation probability will be high if they both photon and spin are parallel to each other [125]. Let us consider the case of the transition presented in figure 3.2; if the photon have a momentum of+¯h (right-handed), there will be excitation of 62.5 % of spin-up electrons at the L3 edge and of 25 % at the L2 edge (L3 and L2 have opposite spin-orbit coupling). If the photon have a momentum of −¯h (left-handed), the probability will be 37.5 % of spin-down electrons at theL3 edge, and 75
% at theL2 edge [109]. Without the spin-orbit coupling of the2pshell, there would be no possible distinction.
The second step of the model is the transition of the excited electron to the spin-split valence band, as we have already seen in the Stoner-Slater model (section 1.2.3). This will act as a detector of the electron polarization. Indeed, if one of the bands is completely filled, then no transition will be possible and there will be a strong dichroism [126].
The XMCD effect is usually calculated as the difference between spectrum recorded with opposite helicities:
IXM CD =IR−IL (3.12)
withIRand IL the intensity corresponding to right and left polarization respectively (red curve in figure 3.2). More than an indication of magnetism, XMCD can be used to obtain information on the orbital and spin moments using the sum rules developed by Tholeet al.[127] and Carra et al. [128].
625 650 675 700 725 750 775
-20
Figure 3.2: X-ray magnetic circular dichroism. On the right side is presented a x-ray absorption spectroscopy (XAS) spectrum recorded with circularly polarized photons at the Fe L-edges (blue curve) and the XMCD spectrum obtained at the same edges (red curve). XAS and XMCD data from Dr. N. Jaouen.,Synchrotron SOLEIL.
3.1.2 Resonant magnetic x-ray scattering
We start again from the same Hamiltonian Htot as used above for the description of ab-sorption:
Htot =Hel+Hrad
Historically, resonant scattering has been treated with the use of a Dirac Hamiltonian by Blume [117]. In this section, we will follow this notation, hence using a Dirac Hamiltonian, but alternative treatments can be found in ref. 129 and 130. The use of this Hamiltonian
introduces the electron spins via Dirac matrices, and also relativistic effects happening to core electrons in heavy atoms [131, 132].
Within the self-consistent field approximation (section 3.1.1), the electronic Hamilto-nianHel is expressed by:
Hel=
N
X
i=1
(cα·pi+βmc2+V(ri)) (3.13) withp the momentum operator,m the rest mass of the electron andc the speed of light.
V(ri) is given in equation 3.3; α and β are the Dirac matrices with complete expressions given by Altarelli in ref. 110. As before, we take into account the interaction between electrons and photons, and thus to replacepi bypi−(e/c)A. Hel is then written as:
Hel =
N
X
i=1
(cα·[pi−(e/c)A] +βmc2+V(ri)) (3.14) As pointed out by Altarelli [110], the soft x-ray range is in the non-relativistic limit of the Dirac Hamiltonian, because the photon energy is very small (500 to 200 eV) compared to the electron rest energy (≈511keV). All the implications are given in Section 15 of ref.
133, but we will keep for our purpose only thatHelcan be re-written under the form [110]:
Hel= with s the electron spin, B and E being the magnetic and electric field of the incident photons.
Using the expression of Hrad given in equation (3.7), the complete Hamiltonian de-scribing the system is: Following the treatment from ref. 108, we can writeHtot as the sum of three Hamilto-nians:
Htot=Hel′ +Hrad′ +Hint′ (3.17) Hel′ andHrad′ are describing the electronic cloud and the electro-magnetic field respectively, andHint′ the interactions between the two systems [117]. Hint′ is expressed as the sum of four independent Hamiltonians:
Hint′ =H1′ +H2′ +H3′ +H4′ (3.18) The complete expressions of H1′, H2′, H3′ and H4′ can be found in ref. 117 and 110. For the following, we only consider the dependency of each term withA, i.e. H1′ andH4′ being quadratic inA, and H2′ and H3′ being linear.
3.1. ABSORPTION AND RESONANT SCATTERING OF X-RAYS 49 Resonant scattering is a second order process (figure 3.1b and c). The transition rate is obtained by extending the Fermi’s Golden Rule to the second order, which is also known as theKramers-Heisenberg equation [134]:
|niis a virtual state in the scattering process with a lifetime ofΓ. The elastic case implies
|ii=|fi =|0i (figure 3.1b), therefore by introducing equation (3.18), equation (3.19) can between|0iand|ni. Equation 3.20 is a general equation which contains both non-resonant (first) and resonant (second) terms. We now review the physical meaning of each of them in the elastic scattering case.
Non resonant terms
H1′ is related to the Thomson scattering (section 3.1.1). We remind to the reader that this phenomenon is related to the scattering of photon by the whole electronic cloud, thus not allowing the difference between core and valence electrons. The latter accounting mainly for the magnetism, Thomson scattering cannot be used as an efficient probe for magnetic properties. It is always observed and is the dominant phenomenon for non-resonant photon energies. In this case and by neglectingH4′, equation (3.20) becomes:
w= 2π
and the scattering cross-section can be written as:
dσ polarization vector of the incident and scattered wave, r0 the Thomson radius. This formula contains theatomic form factor which is defined as
F(q) =X
j
h0|eiq·rj|0i (3.23)
Equation (3.23) is proportional to the Fourier Transform of the electron densityf0: f0 =−r0F(q)(e∗λ·eλ). (3.24) The element H4′ is a small non-resonant term which contains a magnetic contribution related to the spin-orbit interaction. It is smaller compared to the Thomson scattering by a factor (¯hωk/mc2) as demonstrated by Altarelli in ref. 108. Because in the x-ray region
¯
hωk≪mc2, it represents a small contributionfmag to the scattering amplitude. Its exact expression can be found in ref. [135].
Resonant terms
At resonance, the second term of equation (3.20) becomes dominant and then the scattering cross-section can be approximated as:
dσ
dω =|fres|2 (3.25)
with fres given in the electric dipole approximation by the Hannon-Trammel formula [119, 120]: withzthe unit vector of the quantization axis. F1,me is defined as:
F1,me =meX with ml magnetic quantum number defined as −l < ml < l, l the azimuthal quantum number;Γnis the lifetime of the excited state as defined previously.
The Hannon-Trammel formula is describing the transitions between core electrons and d or f shells in transition metals. The magnetic sensitivity arises from the difference F1,1e −F1,−1e which requires different populations in the levelsml = 1 andml = -1.
We can now write the total elastic scattering amplitude as [119, 136]:
f(q,¯hω) =f0(q) +fmag(¯hω) +fres(¯hω) (3.28) withf0 the electron density andfmag the spin-orbit contribution as described above.
Sensitivity to magnetism
Following Hannonet al.[119, 120] we can rewrite fres as:
fres≈fc(e∗λ·eλ)−ifm(e∗λ×eλ)·z (3.29) wherefc and fm are described as complex numbers:
fc =fc′+ifc′′
fm=fm′ +ifm′′
The term fc represents the resonant charge scattering; it gives information about the structural ordering in the sample. Note that this is not Thomson scattering due to the explicit energy dependency.
The second termfmrepresents the resonant magnetization sensitive term that is probed through the transition of a spin-polarized photoelectron from a spin-orbit split core level to a magnetically polarized empty state above the Fermi level. It depends linearly on the magnetization vector and thus provides the magnetic contribution to the scattering amplitude at absorption resonances.
Introducing equation (3.29) in equation (3.28) shows the origin of the sensitivity of resonant scattering to magnetism at resonance due to the presence of the fm term in the expression.